Fulton–Lazarsfeld's connectivity conjecture for symmetric degeneracy loci

Let $X$ be a smooth, projective, connected complex variety of dimension $n$, let $E$ be a vector bundle of rank $e$ on $X$, and let $L$ be a line bundle. Suppose that $E$ has an $L$-valued quadratic form, that is, a section of $S^2E^*\otimes L$. For an integer $k$, let $D_k(E)$ be the subscheme of $X$ where this section has rank at most $k$, and set

$$t(e-k)=\frac{(e-k)(e-k+1)}{2}.$$

Fulton–Lazarsfeld's connectivity conjecture. If

$$\dim D_k(E)=\rho:=n-t(e-k)\geq 1$$

and $S^2E^*\otimes L$ is ample, then $D_k(E)$ is connected.

This is a conjecture attributed to Fulton and Lazarsfeld concerning the connectivity of degeneracy loci of quadratic forms; the supplied source does not state whether it has been resolved.